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Children. (Attentively examining them.) Yes. You have moved that (pointing to it) nearer to the other.
Mother. Very well! Do you observe any other change?
Each of the Children. I see no other,
Mother. What did you notice respecting the oblongs before you turned about?
Children. There were ten placed at equal distances, in a straight line, their long sides turned towards the window, and their short sides towards the door.
Mother. What change did they undergo whilst you turned about?
Children. There were but eight left, and one of them was moved nearer to another. All the rest remained as before.
Mother. Exactly so!
These and many other changes, by diminishing or increasing the number of oblongs, by making the children find out whatever can be observed with respect to their position, &c. are particularly calculated to fix, their attention.
The Mother may vary these first exercises,
Moth. 3 times 1 more once 1 (adding a fourth cube) are 4 times 1.
4 times 1 more once 1 (adding a fifth) are 5 times I.;
Thus she contiuued, till 9 times 1 more once 1 are 10 times 1. The children always repeating after her.
In order to convince herself whether the children thoroughly understood the numbers from 1 to 10, she threw a certain number of cubes upon the table, for instance 5, and asked, How many lie now upon the table?
Childr. 5 are lying upon the table.
Moth. If I add to these 5 one more (adding another to them,,) how many will there be?
Childr. If you add to those five cubes one more, there will be 6.
Childr. 5 times 1 more once 1 are 6 times 1.
Moth. (Taking away the sixth.) Here are 5 again; but (taking up one) if I take away 1 from these five cubes, how many will remain?
Childr. If you take 1 from five, there will remain 4.
After the children have gone through these exercises (which may be varied and exteuded, but very gradually, and always with patience and good humour,) and the elder children practising the younger as far as they know, the Mother may make them count as far as 20 in the same manner, proposing to them similar questions; for instance:
Moth. (Throwing at random a number of cubes, exceeding however ten, upon the table.) How many cubes are here?
Childr. (After having counted them.) There are 13.
Moth. 13 times 1 more once 1, bow many times 1?
Childr. 13 times 1 more once 1, are 14 times 1.
Math. But 13 times 1 less once 1, how many times 1?
Childr. 13 times 1 legs once 1, are twelve times 1.
Moth. If you add to 16 times 1, 3 times I, how many times 1 does it give?
Childr. By adding to 16 times 1, 3 times 1, it will give 19 times 1.
Moth. But if you take from 16 times 1, 4 times 1, how many times 1 will remain?
Childr. By taking from 16 times 1, 4 times 1, 12 times 1 will remain, &c.
As soon as the children were able, with facility, to return corned answers to such questions, with and without the aid of visible objects, the Mother was convinced that they had perfectly acquired the first elements of combining numbers, and she proceeded to the combined unity 2.
Moth. (Placing two cubes together.) Twice 1 are once 2.
Childr. Twice 1 are once 2.
Moth. (Separating them again, and lifting one of them up) 1 is the half of 2.
Childr. 1 is the half of 2.
Moth. (Placing again the second next to the first.) Twice 1 are once 2.
Childr. Twice 1 are once 2.
Moth. (Adding to the two cubes a third, which she placed below them, thus:)
3 times 1 are once 2, and the half of 2. Childr. 3 times 1 are once 2, and the half
Moth. (Adding to the third a fourth cube, so as to form two pair.)
4 times 1 are twice 2.
Childr. 4 times I are twice 2.
Moth. 5 times 1 are twice 2 and the half of 2.
Childr. 5 times 1 are twice 2 and the half of 2.
Moth. 6 times 1 are 3 times 2.
Childr. 6 times 1 are 3 times 2. J/<tf A. 7 times 1 are 3 times 2 and the half of 2.
Childr. 7 times 1 are 3 times 2 and the half of 2.
Moth. 8 times 1 are 4 times 2.
Childr. 8 times 1 are 4 limes 2.
This exercise is carried on to 20 times 1 and 10 times 2, so that twenty cubes are placed by pairs upon the table. This first step in composing, or combining, will require much time and patience.